# unique subgroups

• Oct 18th 2008, 06:57 PM
dori1123
unique subgroups
Prove that $A_4$ has a unique subgroup $V$ of order 4.

I know all 12 elements of $A_4$, and I know $V$ contains ${ 1, (12)(34), (13)(24), (14)(23)}$, but how do I prove that $V$ is a unique subgroup?
• Oct 18th 2008, 06:59 PM
ThePerfectHacker
Quote:

Originally Posted by dori1123
Prove that $A_4$ has a unique subgroup $V$ of order 4.

I know all 12 elements of $A_4$, and I know $V$ contains ${ 1, (12)(34), (13)(24), (14)(23)}$, but how do I prove that $V$ is a unique subgroup?

Since subgroups of order $4$ are Sylow $2$-subgroups it follows all subgroups of order $4$ must be conjugate by Sylow's second theorem. Now $V$ is a normal subgroup of $A_4$. Therefore, it must be the only one.